Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol102.Issue2.1999:
Journal of Computational and Applied Mathematics 102 (1999) 287–302
A family of Pade-type approximants for accelerating
the convergence of sequences
R. Thukral
Pade Research Centre, 39 Deanswood Hill, Leeds, West Yorkshire LS17 5JS, UK
Received 1 September 1997; received in revised form 7 September 1998
Abstract
We describe a collection of Pade-type methods for accelerating the convergence of sequence of functions. The construction and connections of Pade’s methods with other similar methods are given. We examine the e ectiveness of these
new methods, namely integral Pade approximant, modi ed Pade approximant and squared Pade approximant together with
the well-established methods, namely functional Pade approximant and classical Pade approximant, for approximating the
characteristic value and corresponding characteristic function. Estimates of characteristic value and characteristic function
c 1999
derived using integral Pade approximants are found to be substantially more accurate than other similar methods.
Elsevier Science B.V. All rights reserved.
Keywords: Integral Pade approximant; Classical Pade approximant; Functional Pade approximant; Modi ed Pade
approximant; Squared Pade approximant; Integral equation; Neumann series; Convergence acceleration
1. Introduction
In this paper, three new methods for accelerating the convergence of sequence of functions are
introduced and their e ectiveness is examined by determining the characteristic value and the characteristic function of an integral equation. These new methods use functional Pade-type approximants,
where we employ the terminology “Pade-type” as introduced by C. Brezinski, which means that
the denominator polynomial of the rational approximant is arbitrarily prescribed (on the contrary, in
the classical Pade approach the denominator is left free in order to achieve the maximal order of
interpolation). We have introduced the appropriate names for the Pade-type approximants as modi ed Pade approximant, squared Pade approximant and integral Pade approximant. The modi ed
Pade approximant method was the rst modi cation of the classical Pade approximant and this was
further improved by the squared Pade approximant method. Finally, the integral Pade approximant
was developed and this is shown to be a good alternative to functional Pade approximant.
The main reason for developing the appropriate denominators of the new methods was to overcome
the essential diculty encountered by classical Pade approximants and functional Pade approximants.
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
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R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
The major drawback of these methods is the use of the minimal sensitivity principle [1, 2] and
the presence of super
uous zeros in the denominator. Hence, we investigated on the basis that
these new methods should have a similar order and cofactors arrangement in the determinant of
the denominator polynomial as classical Pade approximant and the accuracy of the functional Pade
approximant. Also we use a similar principle of integrating each of the cofactors in the determinant
of the denominator polynomial, which was introduced by Graves-Morris [5, 7, 11] and applied in
functional Pade approximant method.
We describe the fundamentals of the denominator for each of the new Pade-type methods and
the numerator is determined naturally. In order to construct these new methods we use a similar
procedure as classical Pade approximant. The construction of the denominator of the modi ed Pade
approximant method is simply obtained by integrating each of the cofactor in the determinant of the
denominator polynomial of the classical Pade approximant method. This improved method overcomes
the use of the minimal sensitivity principle but it lacks the desired precision and therefore we have
investigated this method further. We construct the denominator of the squared Pade approximant
method by squaring each of the cofactor in the determinant of the denominator polynomial of
the classical Pade approximant method and then we integrate these new cofactors. The precision
was good for some cases but we found that this method is not versatile. Hence, we decided to
construct the denominator of the integral Pade approximant method in a di erent way to those
discussed above and in previously published papers [6–8, 11]. The construction of the denominator
of the integral Pade approximant method is obtained by combining the coecients of the generating
function as cofactors in the determinant of the denominator polynomial. We found that the integral
Pade approximant method is consistent and overcomes all the diculties encountered in the previous
studies [6, 7, 11].
We begin with the generating function f(x; ) of a series of functions given by
f(x; ) =
∞
X
Ci (x) i ;
(1)
i=0
in which Ci (x) ∈ L2 [a; b] are given and [a; b] is the domain of de nition of Ci (x) in some natural
sense. We also suppose that f(x; ) is holomorphic as a function of at the origin = 0. Then
(1) converges for values of || which are small enough. In this paper, we see how the methods of
Pade-type approximation can be used to accelerate the convergence of a series having the form (1).
1.1. The integral Pade approximants method (IPA)
We de ne a rational function r(x; ) to be an integral Pade approximant of type (n; k) for f(x; )
if
r(x; ) = N (x; )=D();
(2)
where N (x; ); D() are polynomials in ; N (x; ) ∈ L2 [a; b] as a function of x and
@{N }6n;
@{D}6k;
D(0) = 1;
N (x; ) − D()f(x; ) = 0(
(3a)
(3b)
n+1
):
(3c)
R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
289
If N (x; ); D() satisfy axiom (3), then there exists a unique r(x; ) de ned by (1). The proofs of
existence and uniqueness are similar to that for the classical Pade approximant [4, 5, 14] and the
rate of convergence is similar to functional Pade approximant. This is evident from this investigation
and in all the other test examples.
For the purpose of this paper, we de ne the denominator polynomial of an integral Pade approximant of type (n; k) as
Z b
Cn+k−1 (x)Cn−k (x) dx
a
Z b
C
(x)Cn−k+1 (x) dx
a n+k−1
D() =
..
.
Z b
Cn+k−1 (x)Cn−1 (x) dx
a
k
Z
b
Cn+k−1 (x)Cn−k+1 (x) dx
···
Cn+k−1 (x)Cn−k+2 (x) dx
..
.
···
a
Z
a
b
Z
a
b
Cn+k−1 (x)Cn (x) dx
k−1
···
···
···
b
Cn+k−1 (x)Cn (x) dx
a
Z b
Cn+k−1 (x)Cn+1 (x) dx
a
;
..
.
Z b
Cn+k−1 (x)Cn+k−1 (x) dx
a
1
Z
(4)
provided D(0) 6= 0 and Ci (x) are the coecients of (1).
The purpose of integrating the elements in (4) is to make the estimates of the characteristic values
independent of the variable x. This is similar to the approach for the functional Pade approximant
in which we overcome the serious problem, use of [1, 2], with the classical Pade approximant
[7, 11]. We take the appropriate roots of denominator polynomial, given by (4), as our estimates of
the characteristic value for the integral Pade approximant.
Naturally, the numerator polynomial N (x; ) follows from (3c) as
N (x; ) = [D()f(x; )]n0 ;
(5)
where this notation, now and in the sequel, indicates that truncation at degree n in has been
e ected. If, in the representation (4), D(0) 6= 0, then r(x; ) de ned by (2), (4) and (5) is an integral
Pade approximant of type (n; k) for f(x; ).
Integral Pade approximants constructed using (3) can be laid out in a table:
(0; 0) (0; 1) (0; 2) · · ·
(1; 0) (1; 1) (1; 2) · · ·
(6)
(2; 0) (2; 1) (2; 2) · · ·
..
..
..
..
.
.
.
.
and this concept is similar to the classical Pade approximants [5].
Conjecture 1.1 (Integral Pade approximant). Let
f(x; ) = N (x; )=D()
(7)
be a meromorphic function with precisely k nite poles. Then; for all n suciently large; there
exists a unique rational function r(x; ) of type (n; k) which interpolates to f(x; ). Hence
Nn (x; )
lim
= f(x; ):
(8)
n→∞ Dn ()
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R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
Summaries of the advantages of integral Pade approximants are as follows:
(i) We do not have to assign a particular value of x in the Neumann series, to obtain an estimate
of the characteristic value using classical Pade approximant [7, 11].
(ii) Integral Pade approximants produces a substantially more accurate estimates than the classical Pade approximant, the modi ed Pade approximant and the other two methods considered
previously, namely Fredholm determinant and Rayleigh–Ritz [7, 11].
(iii) The order of the denominator polynomial of an integral Pade approximant is half the order of
the denominator polynomial of the functional Pade approximant. Therefore, an integral Pade approximant does not possess super
uous zeros, which was a serious problem with the functional
Pade approximant as noted by the investigators [6, 7, 11].
(iv) We do not have to construct a further method, known as the hybrid functional Pade approximant,
to obtain the characteristic function [6, 7, 11].
(v) From the last two advantages it is established that the method of the functional Pade approximants needs much more numerical computation than integral Pade approximants.
(vi) Integral Pade approximant is applicable to a wider class of generating functions. We found a
major drawback of the squared Pade approximant is that the numerical performance of this
method is not suitable when the generating function possesses an alternating or a negative
power series.
(vii) Integral Pade approximant is simpler and more e ective method for obtaining the characteristic
values and the characteristic functions than other similar methods.
1.2. The modi ed Pade approximant method (MPA)
A modi ed Pade approximant of type (n; k) for the given power series (1) is the rational function
r(x; ) = A(x; )=B();
(9)
where A(x; ); B() are polynomials in ; A(x; ) ∈ L2 [a; b] as a function of x and
@{A}6n
(10a)
@{B}6k;
B(0) = 1;
(10b)
A(x; ) − B()f(x; ) = 0(n+1 ):
(10c)
The construction of the denominator polynomial of the modi ed Pade approximant of type (n; k) is
given as
Z b
Cn−k+1 (x) dx
a
Z b
B() = Cn−k+2 (x) dx
a
..
.
k
Z
b
Cn−k+2 (x) dx
···
Cn−k+3 (x) dx
..
.
k−1
···
a
Z
a
b
···
b
Cn+1 (x) dx
a
Z b
Cn+2 (x) dx ;
a
..
.
1
Z
provided B(0) 6= 0 and Ci (x) are the coecients of (1).
(11)
R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
291
Naturally, the numerator polynomial A(x; ) follows from (10c) as
A(x; ) = [B()f(x; )]n0 :
(12)
Each approximant of the sequence of (n; k) type modi ed Pade approximant has precisely k poles. We
take the zeros of (11) as our estimates of the characteristic value for the modi ed Pade approximant.
1.3. The squared Pade approximant method (SPA)
A squared Pade approximant of type (n; k) for the given power series (l) is the rational function
r(x; ) = G(x; )=H ();
(13)
where G(x; ), H () are polynomials in , G(x; ) ∈ L2 [a; b] as a function of x and
@{G}6n;
@{H } 6k;
(14a)
H (0) = 1;
(14b)
G(x; ) − H ()f(x; ) = 0(n+1 ):
(14c)
The construction of the denominator polynomial of the squared Pade approximant of type (n; k) is
given as
Z b
2
(x) dx
Cn−k+1
a
Z
b
2
(x) dx
H () = Cn−k+2
a
..
.
k
Z
b
Z
b
2
(x) dx
Cn−k+2
a
a
2
(x) dx
Cn−k+3
..
.
k−1
Z
b
···
b
···
Z
···
a
a
2
(x) dx
Cn+1
2
Cn+2 (x) dx ;
..
.
1
(15)
provided H (0) 6= 0 and Ci (x) are coecients of (1).
Naturally, the numerator polynomial G(x; ) follows from (14c) as
G(x; ) = [H ()f(x; )]n0 :
(16)
Each approximant of the sequence of (n; k) type Pade approximant has precisely k poles. We actually
take the square root of the zeros formed by (15) as our estimates of the characteristic value.
The outline of this paper is as follows. In Sections 2 and 3 we brie
y describe two well-known
methods, the classical Pade approximant and the functional Pade approximant, respectively. Moreover, in Section 4, we demonstrate the similarity of the three methods, namely, the integral Pade
approximant, the squared Pade approximant and the functional Pade approximant, which produce
identical estimates of the characteristic value. In Section 5 we examine the e ectiveness of these
new methods based on the Pade-type approximants for determining the characteristic values and
the characteristic functions of an integral equation. The technique utilised for solving the integral
equation is based on successive substitution, which is an iterative procedure, yielding a sequence of
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R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
approximations leading to an in nite power series solution. In the process we make two distinct comparisons of the estimates derived using the integral Pade approximant. First, we compare estimates
formed using the row sequence of an integral Pade approximant of type (n; 1) with corresponding
estimates derived from the modi ed Pade approximant of type (n; 1), the squared Pade approximant
of type (n; 1), the functional Pade approximant of type (n; 2) and the classical Pade approximant
of type (n; 1). Then we compare estimates based on another row sequence of an integral Pade approximant of type (n; 2) with corresponding estimates derived from the modi ed Pade approximant
of type (n; 2) the squared Pade approximant of type (n; 2), the functional Pade approximant of type
(n; 4) and the classical Pade approximant of type (n; 2). In Section 6 we illustrate the precision
of a particular characteristic function of integral Pade approximant. The e ectiveness of these new
methods for accelerating the convergence of a sequence of functions was investigated in the context
of the Neumann series of an integral equation. The method of integral Pade approximants proved to
be the most e ective of the methods considered.
2. The classical Pade approximant method (CPA)
A classical Pade approximant of type (n; k) for the given power series (l) is the rational function
r(x; ) = U (x; )=V (x; );
(17)
where U (x; ); V (x; ) are polynomials in ; U (x; ) ∈ L2 [a; b] as a function of x and
@{U }6n;
@{V }6k;
(18a)
V (0) = 1;
(18b)
U (x; ) − V (x; ) f(x; ) = 0(n+k+1 ):
(18c)
The construction of the denominator polynomial of classical Pade approximant of type (n; k) is given
as
Cn−k+1 (x)
Cn−k+2 (x)
V (x; ) =
..
.
k
Cn−k+2 (x) · · · Cn+1 (x)
Cn−k+3 (x) · · · Cn+2 (x)
;
..
..
.
.
k−1
···
1
(19)
provided V (x; 0) 6= 0 and Ci (x) are coecients of (1).
Naturally, the numerator polynomial U (x; ) follows from (18c) as
U (x; ) = [V () f(x; )]n0 :
(20)
Each approximant of the sequence of (n; k)-type Pade approximant has precisely k poles. To determine these zeros, in order to estimate the characteristic value, we must assign a particular value of x
in the Neumann series and this is usually done using the principle of minimal sensitivity [1, 2].
Issues of existence, uniqueness and other related de nitions of classical Pade approximants are
treated in [4, 5] and many other texts.
R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
293
3. The functional Pade approximants method (FPA)
We de ne a rational function r(x; ) to be a functional Pade approximant of type (n; 2k) for f(x; )
if
r(x; ) = p(x; )=q();
(21)
where p(x; ); q() are polynomials in ; p(x; ) ∈ L2 [a; b] as a function of x and
@{p} 6n −
@{q} 62k − 2
)
for ¿0;
(22a)
Z
2
b
q() p(x; ) dx;
a
(22b)
q() = q∗ ();
(22c)
q(0) 6= 0;
(22d)
p(x; ) − q() f(x; ) = 0 (n+1 ):
(22e)
The asterisk in (22c) denotes the functional complex conjugate.
If p(x; ), q(), satisfy (22a)–(e), then r(x; ) de ned by (1) is unique; the questions of existence,
uniqueness and degeneracy are treated in [9]. The explicit formula for the denominator polynomial
is given by
0
M01
−M01
0
..
.
..
q() =
.
−M
−M
0; 2k−1
1; 2k−1
2k
2k−1
· · · M0; 2k−1 M0; 2k
· · · M1; 2k−1 M1; 2k
..
..
..
:
.
.
.
···
0
M2k−1; 2k
···
1
(23)
The elements of (23) are de ned by
Mij =
j−i−1 Z b
X
i=0
a
Cl+i+n−2k+1 (x)[Cj−l+n−2k (x)]∗ dx;
(24)
for i = 0; 1; : : : ; 2k and j = i + 1; i + 2; : : : ; 2k and taking Cj (x) := 0 if j ¡ 0:
We know that the polynomials produced by (23) are strictly positive for ∈ R and their zeros
occur in complex conjugate pairs close to the real axis [6, 7, 11]. Ideally, we take the real parts of
the zeros of q() as our estimates of the characteristic values c .
The numerator polynomial p(x; ) follows from (22e) as
p(x; ) = [q() f(x; )]n0 :
(25)
If, in the representation (23), q(0) 6= 0, then r(x; ) de ned by (21), (23) and (25) is the functional
Pade approximant of type (n; 2k) for f(x; ).
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R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
In addition, the denominator polynomial of a hybrid functional Pade approximant is de ned in
terms of the roots of the denominator of the corresponding functional Pade approximant [6, 7, 11].
We actually take the real parts of the roots of the functional Pade approximant and express the
hybrid functional Pade approximant of type (n; k) as
qH () =
k
Y
i=1
( − iR ):
(26)
The associated numerator polynomial is de ned as
pH (x; ) = [f(x; ) qH ()]n0 :
(27)
Since n and k govern the degree of the numerator and denominator, respectively, we express the
hybrid functional Pade approximant of type (n; k) as
r H (x; ) = pH (x; )=qH ():
(28)
We nd that (28) is identical to the integral Pade approximant (2) for k = 1. It is obvious that
the hybrid functional Pade approximant is dependent on functional Pade approximant and therefore
requires much more numerical computation than the integral Pade approximant. We can easily prove
this by comparing the dimensions of the determinant and the degree of the denominator polynomial
of functional Pade approximant and integral Pade approximant, which are given by (23) and (4),
respectively.
4. Equivalence of the estimates
Here we shall observe how three methods, namely integral Pade approximant, the squared Pade approximant and the functional Pade approximant produce similar estimates of the characteristic value.
We shall do this by showing that the limit of the poles of the respective denominator polynomial
satis es identical equations.
First we begin by expanding (4), the denominator polynomial of the integral Pade approximant
of type (n; 1):
Z b
C (x) Cn−1 (x) dx
D() = a n
Z
b
Cn2 (x) dx
a
1
which gives
D() =
Z
a
b
Cn (x) Cn−1 (x) dx −
Z
a
;
b
Cn2 (x) dx:
(29)
The characteristic value of the integral Pade approximant of type (n; 1) is calculated by solving (29)
and thus we have
=
Z
a
b
Cn (x)Cn−1 (x) dx
,Z
a
b
Cn2 (x) dx:
(30)
R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
295
Similarly, expanding the denominator polynomial of the functional Pade approximant of type (n; 2),
0
Z b
q() =
2
(x) dx
Cn−1
−
a
2
b
Z
2
(x) dx
Cn−1
a
2
b
Z
Cn−1 (x)Cn (x) dx
a
Z
0
a
b
;
Cn2 (x) dx
1
which gives
Z
q() =
b
2
Cn−1
(x) dx
a
−2
Z
a
"Z
b
2
Cn−1
(x) dx
a
b
2
(x) dx
Cn−1
"Z
+
2
b
Z
a
Cn2 (x) dx
#
#
b
Cn (x)Cn−1 (x) dx :
a
(31)
Solving (31) by a standard quadratic formula and simplifying gives rise to
=
Rb
[ a Cn (x)Cn−1 (x) dx ±
q R
b
[{ a Cn (x)Cn−1 (x) dx}2 −
Rb
[ a Cn2 (x) dx]
Rb
a
Rb
2
Cn2 (x) dx a Cn−1
(x) dx]]
:
(32)
We know from previous studies [6, 7, 11] that (32) produces a pair of complex conjugates which
are close to the real axis, that is
"Z
b
Cn (x)Cn−1 (x) dx
a
#2
¡
"Z
b
a
Cn2 (x) dx
Z
a
b
2
Cn−1
(x) dx
#
(33)
and
lim
n→∞
"Z
b
Cn (x)Cn−1 (x) dx
a
#2
"Z
=
a
b
Cn2 (x) dx
Z
a
b
2
(x) dx
Cn−1
#
:
(34)
Therefore, the estimate of the characteristic value of the functional Pade approximant of type (n; 2)
method is given by
=
"Z
a
b
Cn (x)Cn−1 (x) dx
# ,"Z
a
b
#
Cn2 (x) dx :
(35)
Alternatively, we can nd the estimates of the functional Pade approximant by di erentiating (31)
w.r.t. and with a convenient normalisation we obtain (35) [8, 11].
First we begin by expanding (15), the denominator polynomial of the squared Pade approximant
of type (n; 1)
Z b
Z b
2
2
(x)
dx
C
(x)
dx
C
n
n−1
;
H () = a
a
1
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R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
which gives
Z
H () =
b
2
(x) dx
Cn−1
a
−
Z
a
b
Cn2 (x) dx:
(36)
The characteristic value of squared Pade approximant of type (n; 1) is based on solving (36); thus
we have
=
Z
b
2
(x) dx
Cn−1
a
,Z
a
b
Cn2 (x) dx:
(37)
If we rearrange (34), we have
lim
n→∞
"Z
a
b
2
(x) dx
Cn−1
#
=
Rb
[ a Cn (x)Cn−1 (x) dx]2
Rb
[ a Cn2 (x) dx]
(38)
and substituting the right-hand side of (38) into (37) we obtain
=
"R b
a
Cn (x)Cn−1 (x) dx
Rb
a
Cn2 (x) dx
#2
:
(39)
It has been established in [5, 6, 8, 10] that the zeros, that is characteristic value, of the functional
Pade approximant converges as
lim n = 2 :
n →∞
(40)
This result also applies to the squared Pade approximant.
However, we actually take the square root of (37) as our estimates of the characteristic value
v
,Z
u Z b
b
u
2
t
= Cn−1 (x) dx
Cn2 (x) dx :
a
a
(41)
Therefore, we expect that, for all n suciently large, (41) is equivalent to (30). It should be
noted that this is true mathematically, but we nd a serious problem numerically that is when the
generating function possesses an alternating or a negative power series.
Mathematically, we have demonstrated the fact that these three methods produce a similar accuracy
for the estimates of the characteristic value, that is (30), (35) and (41) are similar for the rst row
sequence. We conjecture that these three methods produce identical estimates for any type of row
sequence. However, numerically we nd that the integral Pade approximant has better precision than
the squared Pade approximant and the functional Pade approximant. This is empirically evident from
Table 2.
5. Application to an integral equation
To determine the consistency of these new methods, we actually tested them on a previous
investigation [11] and further examples were taken from Moiseiwitsch [12]. These ndings are
R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
297
generalised by illustrating the e ectiveness of these new methods for determining the characteristic
values and the characteristic function of a familiar linear integral equation in the following example.
We investigate the convergence of sequences of these new methods for the Neumann series
solution of the linear integral equation
f(x; ) = 1 +
Z
1
k(x; y)f(y; ) dy;
(42)
0
where
k(x; y) =
1 + x − y;
1 + y − x;
0 6 y 6 x 6 1;
0 6 x 6 y 6 1:
This integral equation is a Fredholm of the second kind with a nondegenerate kernel and has been
previously considered by Graves–Morris [7] and Coope and Graves–Morris [6].
The characteristic functions of this equation can be found by converting it to a second-order
ordinary di erential equation [3]. The explicit solution of (42) is
2 cosh (x − 1=2)
f(x; ) =
;
(43)
2 cosh =2 − 3 sinh =2
√
where = 2. The denominator of (43) is analytic as a function of and has just one simple zero
at c = 1:22290658 : : :, corresponding to a single characteristic value c = 0:7477502556 : : :.
It is familiar that the Neumann series of (42) converges for ||¡c [12]. The rst few terms of
this series are
"
"
#
#
∞
X
161 5
5
1 2 1
1 4 2
1 2
i
f(x; ) =
x−
x−
+
+ ···;
+
Ci (x) = 1 +
+ x−
+
4
2
96
4
2
6
2
i=0
(44)
as may be found by iteration of (42).
We make two distinct comparisons of the estimates derived using the integral Pade approximants.
First, we compare estimates formed using the row sequence of the integral Pade approximant of
type (n; 1) with corresponding estimates derived from the classical Pade approximant of type (n; 1),
the modi ed Pade approximant of type (n; 1), the squared Pade approximant of type (n; 1) and
the functional Pade approximant of type (n; 2). The results in Table 1 are the estimates of the
characteristic value for each of the ve iterative methods described and we nd that the estimates
from the integral Pade approximant gives better approximations that the classical Pade approximant,
the modi ed Pade approximant and it is similar to the functional Pade approximant and the squared
Pade approximant. The results in Table 2 are the estimates of the characteristic value, but showing
results derived from another row sequence of the integral Pade approximant of type (n; 2) and with
corresponding estimates derived from the classical Pade approximant of type (n; 2), the modi ed
Pade approximant of type (n; 2), the squared Pade approximant of type (n; 2) and the functional
Pade approximant of type (n; 4). We see that the row sequence of the integral Pade approximants
gives better approximation than the other methods, including the functional Pade approximants. In
each case, the comparisons with other methods were made using a similar amount of data, which
is, using a similar number of terms of (44).
The proof that the row sequence of Pade approximant estimates shown in Table 1 converge geometrically to the characteristic value follows as a consequence of Sa ’s [14] extension of Montessus’
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R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
Table 1
Estimates showing the precision of the characteristic value c derived using the ve methods described
n
IPA = FPA
ca
SPA
c
MPA
c
CPA
c
1
2
3
4
5
0.74766
0.74775011
0.7477502553
0.7477502555635
0.7477502555638444
0.7488
0.747752
0.7477502583
0.747750255568
0.747750255563853
0.75
0.74766
0.747754
0.74775011
0.747750261
0.8
0.745
0.74785
0.747746
0.74775043
a
The exact value of c = 0:747750255563845043 : : : .
Table 2
Estimates showing the precision of the characteristic value c derived using the ve methods described
IPA
n ca
FPA
c
SPA
c
MPA
c
CPA
c
1
2
3
4
0.7453
0.74775025556262
0.74775025556384496
0.7477502555638450433815
0.747755
0.7477502557
0.747750255563858
0.747750255563845045
0.74767
0.7477504
0.7477502545
0.74775025557
0.7486
0.74774
0.7477504
0.74775025
0.7475011
0.74775025556375
0.747750255563845035
0.7477502555638450433881
a
The exact value of c = 0:74775025556384504338894 : : : .
theorem [13]. Likewise, the proof that the functional Pade approximant estimates shown in the Table
1 converge geometrically to the characteristic value, follows as a consequence of the row convergence theorem of Graves–Morris and Sa [10]. All these mathematical results are empirically evident
from Tables 1 and 2. It is also clear that the integral Pade approximant method converges much
faster than the other methods. This is due to the standard theory of Fredholm integral equations of
the second kind [12, 15], that is expressed by
f(x; ) = g(x) +
Z
b
k(x; y)f(y; ) dy;
(45)
a
it is known that when g(x) ∈ L2 [a; b] and k(x; y) is an L2 kernel,
f(x; y) =
P(x; )
;
Q()
(46)
where
(i) P(x; ) is holomorphic as a function of ,
(ii) P(x; ) ∈ L2 [a; b] for a xed ,
(iii) Q() is holomorphic function of ,
(iv) Q(0) 6= 0.
We have noted that the more serious practical diculty with the classical Pade approximant method
is that the estimates of the characteristic values of depend on the value of x selected contrary
to the exact result (46). If we use the integral Pade approximants for accelerating convergence of
the Neumann series, the estimates of the characteristic values are independent of the variable x,
as expressed in (46). This observation goes some way towards explaining the remarkable precision
299
R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
of estimates of the characteristic values derived from the integral Pade approximants as shown in
Tables 1 and 2.
6. Precision of the approximate solution
In Fig. 1 we display the exact (analytic) solution and its approximations obtained using the integral Pade approximant of type (2; 1), the modi ed Pade approximant of type (2; 1), the squared
Pade approximant of type (2; 1) and the functional Pade approximant of type (2; 2). Also in
Fig. 1 we see a remarkable precision of the integral Pade approximant, where graphically there
is no signi cant di erence between the exact and the integral Pade approximant. Accordingly, in
Table 3, we show the errors incurred by the integral Pade approximant, the modi ed Pade approximant, the squared Pade approximant and the functional Pade approximant methods for x = 0(0:1)0:5
in the solution of (42). We list the appropriate rational functions displayed.
Solution of the integral equation (42) using the integral Pade approximant of type (2; 1) is
N (x; ) 1 + x2 − x +
r(x; ) =
=
D()
4399
27045
+
1 4
x
6
− 13 x3 +
72337
1 − 54090
4399 2
x
27045
+
217
x
54090
−
217
36060
2
:
(47)
Fig. 1. The analytic solution (exact) of (42) for = 1. The curves IPA and MPA are indistinguishable from the exact
curve. Whereas the curve FPA is perceptibly di erent and SPA is beyond the exact curve.
Table 3
Errors occurring in the solution of (42) using the integral Pade approximant, the modi ed Pade approximant, the squared
Pade approximant and the functional Pade approximant method
x
IPA
=1
MPA
=1
SPA
=1
FPA
=1
0
0.1
0.2
0.3
0.4
0.5
−0.00088
−0.00066
−0.00021
0.00031
0.00070
0.00085
−0.0036
−0.0032
−0.0026
−0.0020
−0.0016
−0.0014
−3.39
−3.19
−3.04
−2.93
−2.86
−2.84
0.069
0.050
0.013
−0.026
−0.053
−0.063
300
R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
Solution of (42) based on the squared Pade approximant of type (2; 1) is
G(x; ) 1 + x2 − x +
=
r(x; ) =
H ()
2917
101115
+
1 4
x
6
− 13 x3 −
72337
1 − 40446
2917 2
x
101115
+
18409
x
40446
−
18409
26964
2
:
(48)
Solution of (42) based on the functional Pade approximant of type (2; 2) is
p(x; ) 1 + x2 − x −
=
r(x; ) =
q()
5279
4494
x2 −
+ 16 x4 − 13 x3 − 5279
4494
1 − 18030
+ 72337
2
6741
40446
1507
x
40446
−
9041
40446
2
:
(49)
Solution of (42) based on the modi ed Pade approximant of type (2; 1) is
A(x; ) 1 + x2 − x +
=
r(x; ) =
B()
13
80
+
1 4
x
6
− 13 x3 +
1 − 107
80
13 2
x
80
+
1
x
15
−
1
10
2
:
(50)
The rational functions above were suitably normalised.
It has been established in previous studies [6, 7, 11] that the estimates of characteristic function
based on the functional Pade approximants are inferior to those from the hybrid functional Pade
approximants. The reason for the poor performance of the functional Pade approximants for the
estimation of the characteristic function is that the denominator polynomials (11) possess super
uous
zeros, and thus the hybrid functional Pade approximant method was introduced. We do not use the
hybrid functional Pade approximant, not because it produces identical results to the integral Pade
approximant, but because of much more numerical computation involved in deriving it. Furthermore,
we have found that the precision of the characteristic function for the squared Pade approximant
method is considerably smaller than the integral Pade approximant. Finally, although the approximate
solution for the characteristic function for the modi ed Pade approximant is satisfactory, it lacks the
precision of the integral Pade approximant.
7. Remarks and conclusion
Three new methods for producing a sequence of rational approximations to the solution of a linear
integral equation have been described and their e ectiveness has been investigated in many examples.
These new methods are essentially for accelerating the convergence of a sequence of functions. In
the context of a familiar linear integral equation, the method of integral Pade approximants is shown
to be much more ecient in calculating the characteristic value and substantially more accurate for
calculating the approximate solution than other similar techniques.
The purpose of demonstrating the integral Pade approximants method for two types of row sequence is to illustrate the accuracy of the approximate solution, the stability of the convergence and
the consistency of the method. Furthermore, we have demonstrated that this new method is much
more ecient and does not have the drawbacks of the functional Pade approximants and the classical Pade approximants. From this illustrated example and in all other test examples, it is clear that
the hybrid functional Pade approximants method is inferior to the integral Pade approximants for
k ¿ 2, because of the accuracy of the characteristic values of the functional Pade approximants is
less than the integral Pade approximants method. Finally, an analytical investigation of the integral
Pade approximant is a subject of further research.
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R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
Acknowledgements
I am greatly indebted to an anonymous referee and Professor D.B. Ingham for helpful comments
on this paper.
Appendix. Application to another integral equation
We illustrate the e ectiveness of new methods by taking another well known example. We investigate the convergence of sequences of integral Pade approximants for the Neumann series solution
of the linear integral equation
f(x; ) = 1 +
Z
1
k(x; y)f(y) dy;
(A.1)
−1
where
k(x; y) =
(
8−1 (1 + y)(1 − x); −16y6x61;
8−1 (1 + x)(1 − y); −16x6y61:
This integral equation is a Fredholm of the secod kind with a nondegenerate kernel previously
considered by Graves-Morris and Thukral [11].
The characteristic functions of this equation can be found by converting it to a second-order
ordinary di erential equation [3]. The explicit solution of (1) is
f(x; s ) = sin[2−1 s(1 + x)];
s ∈ ℵ;
Table 4
Estimates showing the precision of the characteristic value 1 derived using the ve methods described
n
IPA= FPA
1a
SPA
1
MPA
1
CPA
1
1
2
3
4
10
9.871
9.86962
9.8696046
10.95
9.877
9.86969
9.8696054
12
10
9.88
9.871
8
9.6
9.84
9.866
a
The exact value of 1 = 2 = 9:8696044010 : : : .
Table 5
Estimates showing the precision of the characteristic value 1 derived using the ve methods described
n
IPA
1a
FPA
1
SPA
1
MPA
1
CPA
1
1
2
3
4
5
9.871
9.8696046
9.8696044013
9.8696044010896
9.8696044010893593
9.24
9.869609
9.8696044025
9.869604401091
9.869604401089361
10.03
9.86964
9.86960444
9.86960440115
9.86960440108946
10.25
9.8751
9.86975
9.869609
9.8696046
10.14
9.878
9.8699
9.86962
9.8696051
a
The exact value of 1 = 2 = 9:8696044010893586188 : : : .
302
R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
Table 6
Estimates showing the precision of the characteristic value 3 derived using the three methods described
n
IPA
3a
FPA
3
SPA
3
MPA
3
CPA
3
1
2
3
4
5
—
90.1
88.97
88.845
88.8289
—
104.5
90.26
88.999
88.848
—
126.1
90.99
89.07
88.858
—
170.1
102.1
92.6
90.1
—
64.79
78.31
84.52
87.15
a
The exact value of 3 = 92 = 88:8264396 : : : .
with corresponding characteristic value
s = (s)2 ;
s ∈ ℵ:
References
[1] C. Alabiso, P. Butera, G.M. Prosperi, Resolvent operator, Pade approximation and bound states in potential scattering,
Nuovo Cim. 3 (1970) 831–839.
[2] C. Alabiso, P. Butera, G.M. Prosperi, Variational principles and Pade approximants. Bound states in potential theory,
Nucl. Phys. B 31 (1971) 141–162.
[3] C.T.H. Baker, The Numerical Treatment of Integral Equations, Oxford University Press, Oxford, 1978.
[4] G.A. Baker, Essentials of Pade approximants, Academic Press, New York, 1975.
[5] G.A. Baker, P.R. Graves-Morris, Pade approximants, Cambridge University Press, Cambridge, UK, 1996.
[6] I.D. Coope, P.R. Graves-Morris, The rise and fall of the vector epsilon algorithm, Numer. Algorithm 5 (1993)
275–286.
[7] P.R. Graves-Morris, Solution of integral equations using generalised inverse, function-valued Pade approximants I.
J. Comp. Appl. Math. 32 (1990) 117–124.
[8] P.R. Graves-Morris, A review of Pade methods for the acceleration of convergence of a sequence of vectors. Appl.
Numer. Math. 15 (1994) 153–174.
[9] P.R. Graves-Morris, C.D. Jenkins, Degeneracies of generalised inverse, vector-valued Pade approximants, Constr.
Approx. 5 (1989) 463– 485.
[10] P.R. Graves-Morris, E.B. Sa , Row convergence theorems for generalised inverse, vector-valued Pade approximants,
J. Comput. Appl. 23 (1988) 63–85.
[11] P.R. Graves-Morris, R. Thukral, Solution of integral equations using function-valued Pade approximants II, Numer.
Algorithms 3 (1992) 223–234.
[12] B.L. Moiseiwitsch, Integral Equations, Longman, New York, 1977.
[13] R. de Montessus de Ballore, Sur les fractions continues algebriques, Bull. Soc. Math. France 30 (1902) 28–36.
[14] E.B. Sa , An extension of R. de Montessus de Ballore’s theorem on the convergence of interpolating rational
functions, J. Approx. Theory 6 (1972) 63– 67.
[15] F. Smithies, Integral Equations, Cambridge University Press, Cambridge, 1962.
A family of Pade-type approximants for accelerating
the convergence of sequences
R. Thukral
Pade Research Centre, 39 Deanswood Hill, Leeds, West Yorkshire LS17 5JS, UK
Received 1 September 1997; received in revised form 7 September 1998
Abstract
We describe a collection of Pade-type methods for accelerating the convergence of sequence of functions. The construction and connections of Pade’s methods with other similar methods are given. We examine the e ectiveness of these
new methods, namely integral Pade approximant, modi ed Pade approximant and squared Pade approximant together with
the well-established methods, namely functional Pade approximant and classical Pade approximant, for approximating the
characteristic value and corresponding characteristic function. Estimates of characteristic value and characteristic function
c 1999
derived using integral Pade approximants are found to be substantially more accurate than other similar methods.
Elsevier Science B.V. All rights reserved.
Keywords: Integral Pade approximant; Classical Pade approximant; Functional Pade approximant; Modi ed Pade
approximant; Squared Pade approximant; Integral equation; Neumann series; Convergence acceleration
1. Introduction
In this paper, three new methods for accelerating the convergence of sequence of functions are
introduced and their e ectiveness is examined by determining the characteristic value and the characteristic function of an integral equation. These new methods use functional Pade-type approximants,
where we employ the terminology “Pade-type” as introduced by C. Brezinski, which means that
the denominator polynomial of the rational approximant is arbitrarily prescribed (on the contrary, in
the classical Pade approach the denominator is left free in order to achieve the maximal order of
interpolation). We have introduced the appropriate names for the Pade-type approximants as modi ed Pade approximant, squared Pade approximant and integral Pade approximant. The modi ed
Pade approximant method was the rst modi cation of the classical Pade approximant and this was
further improved by the squared Pade approximant method. Finally, the integral Pade approximant
was developed and this is shown to be a good alternative to functional Pade approximant.
The main reason for developing the appropriate denominators of the new methods was to overcome
the essential diculty encountered by classical Pade approximants and functional Pade approximants.
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 2 2 9 - 5
288
R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
The major drawback of these methods is the use of the minimal sensitivity principle [1, 2] and
the presence of super
uous zeros in the denominator. Hence, we investigated on the basis that
these new methods should have a similar order and cofactors arrangement in the determinant of
the denominator polynomial as classical Pade approximant and the accuracy of the functional Pade
approximant. Also we use a similar principle of integrating each of the cofactors in the determinant
of the denominator polynomial, which was introduced by Graves-Morris [5, 7, 11] and applied in
functional Pade approximant method.
We describe the fundamentals of the denominator for each of the new Pade-type methods and
the numerator is determined naturally. In order to construct these new methods we use a similar
procedure as classical Pade approximant. The construction of the denominator of the modi ed Pade
approximant method is simply obtained by integrating each of the cofactor in the determinant of the
denominator polynomial of the classical Pade approximant method. This improved method overcomes
the use of the minimal sensitivity principle but it lacks the desired precision and therefore we have
investigated this method further. We construct the denominator of the squared Pade approximant
method by squaring each of the cofactor in the determinant of the denominator polynomial of
the classical Pade approximant method and then we integrate these new cofactors. The precision
was good for some cases but we found that this method is not versatile. Hence, we decided to
construct the denominator of the integral Pade approximant method in a di erent way to those
discussed above and in previously published papers [6–8, 11]. The construction of the denominator
of the integral Pade approximant method is obtained by combining the coecients of the generating
function as cofactors in the determinant of the denominator polynomial. We found that the integral
Pade approximant method is consistent and overcomes all the diculties encountered in the previous
studies [6, 7, 11].
We begin with the generating function f(x; ) of a series of functions given by
f(x; ) =
∞
X
Ci (x) i ;
(1)
i=0
in which Ci (x) ∈ L2 [a; b] are given and [a; b] is the domain of de nition of Ci (x) in some natural
sense. We also suppose that f(x; ) is holomorphic as a function of at the origin = 0. Then
(1) converges for values of || which are small enough. In this paper, we see how the methods of
Pade-type approximation can be used to accelerate the convergence of a series having the form (1).
1.1. The integral Pade approximants method (IPA)
We de ne a rational function r(x; ) to be an integral Pade approximant of type (n; k) for f(x; )
if
r(x; ) = N (x; )=D();
(2)
where N (x; ); D() are polynomials in ; N (x; ) ∈ L2 [a; b] as a function of x and
@{N }6n;
@{D}6k;
D(0) = 1;
N (x; ) − D()f(x; ) = 0(
(3a)
(3b)
n+1
):
(3c)
R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
289
If N (x; ); D() satisfy axiom (3), then there exists a unique r(x; ) de ned by (1). The proofs of
existence and uniqueness are similar to that for the classical Pade approximant [4, 5, 14] and the
rate of convergence is similar to functional Pade approximant. This is evident from this investigation
and in all the other test examples.
For the purpose of this paper, we de ne the denominator polynomial of an integral Pade approximant of type (n; k) as
Z b
Cn+k−1 (x)Cn−k (x) dx
a
Z b
C
(x)Cn−k+1 (x) dx
a n+k−1
D() =
..
.
Z b
Cn+k−1 (x)Cn−1 (x) dx
a
k
Z
b
Cn+k−1 (x)Cn−k+1 (x) dx
···
Cn+k−1 (x)Cn−k+2 (x) dx
..
.
···
a
Z
a
b
Z
a
b
Cn+k−1 (x)Cn (x) dx
k−1
···
···
···
b
Cn+k−1 (x)Cn (x) dx
a
Z b
Cn+k−1 (x)Cn+1 (x) dx
a
;
..
.
Z b
Cn+k−1 (x)Cn+k−1 (x) dx
a
1
Z
(4)
provided D(0) 6= 0 and Ci (x) are the coecients of (1).
The purpose of integrating the elements in (4) is to make the estimates of the characteristic values
independent of the variable x. This is similar to the approach for the functional Pade approximant
in which we overcome the serious problem, use of [1, 2], with the classical Pade approximant
[7, 11]. We take the appropriate roots of denominator polynomial, given by (4), as our estimates of
the characteristic value for the integral Pade approximant.
Naturally, the numerator polynomial N (x; ) follows from (3c) as
N (x; ) = [D()f(x; )]n0 ;
(5)
where this notation, now and in the sequel, indicates that truncation at degree n in has been
e ected. If, in the representation (4), D(0) 6= 0, then r(x; ) de ned by (2), (4) and (5) is an integral
Pade approximant of type (n; k) for f(x; ).
Integral Pade approximants constructed using (3) can be laid out in a table:
(0; 0) (0; 1) (0; 2) · · ·
(1; 0) (1; 1) (1; 2) · · ·
(6)
(2; 0) (2; 1) (2; 2) · · ·
..
..
..
..
.
.
.
.
and this concept is similar to the classical Pade approximants [5].
Conjecture 1.1 (Integral Pade approximant). Let
f(x; ) = N (x; )=D()
(7)
be a meromorphic function with precisely k nite poles. Then; for all n suciently large; there
exists a unique rational function r(x; ) of type (n; k) which interpolates to f(x; ). Hence
Nn (x; )
lim
= f(x; ):
(8)
n→∞ Dn ()
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R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
Summaries of the advantages of integral Pade approximants are as follows:
(i) We do not have to assign a particular value of x in the Neumann series, to obtain an estimate
of the characteristic value using classical Pade approximant [7, 11].
(ii) Integral Pade approximants produces a substantially more accurate estimates than the classical Pade approximant, the modi ed Pade approximant and the other two methods considered
previously, namely Fredholm determinant and Rayleigh–Ritz [7, 11].
(iii) The order of the denominator polynomial of an integral Pade approximant is half the order of
the denominator polynomial of the functional Pade approximant. Therefore, an integral Pade approximant does not possess super
uous zeros, which was a serious problem with the functional
Pade approximant as noted by the investigators [6, 7, 11].
(iv) We do not have to construct a further method, known as the hybrid functional Pade approximant,
to obtain the characteristic function [6, 7, 11].
(v) From the last two advantages it is established that the method of the functional Pade approximants needs much more numerical computation than integral Pade approximants.
(vi) Integral Pade approximant is applicable to a wider class of generating functions. We found a
major drawback of the squared Pade approximant is that the numerical performance of this
method is not suitable when the generating function possesses an alternating or a negative
power series.
(vii) Integral Pade approximant is simpler and more e ective method for obtaining the characteristic
values and the characteristic functions than other similar methods.
1.2. The modi ed Pade approximant method (MPA)
A modi ed Pade approximant of type (n; k) for the given power series (1) is the rational function
r(x; ) = A(x; )=B();
(9)
where A(x; ); B() are polynomials in ; A(x; ) ∈ L2 [a; b] as a function of x and
@{A}6n
(10a)
@{B}6k;
B(0) = 1;
(10b)
A(x; ) − B()f(x; ) = 0(n+1 ):
(10c)
The construction of the denominator polynomial of the modi ed Pade approximant of type (n; k) is
given as
Z b
Cn−k+1 (x) dx
a
Z b
B() = Cn−k+2 (x) dx
a
..
.
k
Z
b
Cn−k+2 (x) dx
···
Cn−k+3 (x) dx
..
.
k−1
···
a
Z
a
b
···
b
Cn+1 (x) dx
a
Z b
Cn+2 (x) dx ;
a
..
.
1
Z
provided B(0) 6= 0 and Ci (x) are the coecients of (1).
(11)
R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
291
Naturally, the numerator polynomial A(x; ) follows from (10c) as
A(x; ) = [B()f(x; )]n0 :
(12)
Each approximant of the sequence of (n; k) type modi ed Pade approximant has precisely k poles. We
take the zeros of (11) as our estimates of the characteristic value for the modi ed Pade approximant.
1.3. The squared Pade approximant method (SPA)
A squared Pade approximant of type (n; k) for the given power series (l) is the rational function
r(x; ) = G(x; )=H ();
(13)
where G(x; ), H () are polynomials in , G(x; ) ∈ L2 [a; b] as a function of x and
@{G}6n;
@{H } 6k;
(14a)
H (0) = 1;
(14b)
G(x; ) − H ()f(x; ) = 0(n+1 ):
(14c)
The construction of the denominator polynomial of the squared Pade approximant of type (n; k) is
given as
Z b
2
(x) dx
Cn−k+1
a
Z
b
2
(x) dx
H () = Cn−k+2
a
..
.
k
Z
b
Z
b
2
(x) dx
Cn−k+2
a
a
2
(x) dx
Cn−k+3
..
.
k−1
Z
b
···
b
···
Z
···
a
a
2
(x) dx
Cn+1
2
Cn+2 (x) dx ;
..
.
1
(15)
provided H (0) 6= 0 and Ci (x) are coecients of (1).
Naturally, the numerator polynomial G(x; ) follows from (14c) as
G(x; ) = [H ()f(x; )]n0 :
(16)
Each approximant of the sequence of (n; k) type Pade approximant has precisely k poles. We actually
take the square root of the zeros formed by (15) as our estimates of the characteristic value.
The outline of this paper is as follows. In Sections 2 and 3 we brie
y describe two well-known
methods, the classical Pade approximant and the functional Pade approximant, respectively. Moreover, in Section 4, we demonstrate the similarity of the three methods, namely, the integral Pade
approximant, the squared Pade approximant and the functional Pade approximant, which produce
identical estimates of the characteristic value. In Section 5 we examine the e ectiveness of these
new methods based on the Pade-type approximants for determining the characteristic values and
the characteristic functions of an integral equation. The technique utilised for solving the integral
equation is based on successive substitution, which is an iterative procedure, yielding a sequence of
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R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
approximations leading to an in nite power series solution. In the process we make two distinct comparisons of the estimates derived using the integral Pade approximant. First, we compare estimates
formed using the row sequence of an integral Pade approximant of type (n; 1) with corresponding
estimates derived from the modi ed Pade approximant of type (n; 1), the squared Pade approximant
of type (n; 1), the functional Pade approximant of type (n; 2) and the classical Pade approximant
of type (n; 1). Then we compare estimates based on another row sequence of an integral Pade approximant of type (n; 2) with corresponding estimates derived from the modi ed Pade approximant
of type (n; 2) the squared Pade approximant of type (n; 2), the functional Pade approximant of type
(n; 4) and the classical Pade approximant of type (n; 2). In Section 6 we illustrate the precision
of a particular characteristic function of integral Pade approximant. The e ectiveness of these new
methods for accelerating the convergence of a sequence of functions was investigated in the context
of the Neumann series of an integral equation. The method of integral Pade approximants proved to
be the most e ective of the methods considered.
2. The classical Pade approximant method (CPA)
A classical Pade approximant of type (n; k) for the given power series (l) is the rational function
r(x; ) = U (x; )=V (x; );
(17)
where U (x; ); V (x; ) are polynomials in ; U (x; ) ∈ L2 [a; b] as a function of x and
@{U }6n;
@{V }6k;
(18a)
V (0) = 1;
(18b)
U (x; ) − V (x; ) f(x; ) = 0(n+k+1 ):
(18c)
The construction of the denominator polynomial of classical Pade approximant of type (n; k) is given
as
Cn−k+1 (x)
Cn−k+2 (x)
V (x; ) =
..
.
k
Cn−k+2 (x) · · · Cn+1 (x)
Cn−k+3 (x) · · · Cn+2 (x)
;
..
..
.
.
k−1
···
1
(19)
provided V (x; 0) 6= 0 and Ci (x) are coecients of (1).
Naturally, the numerator polynomial U (x; ) follows from (18c) as
U (x; ) = [V () f(x; )]n0 :
(20)
Each approximant of the sequence of (n; k)-type Pade approximant has precisely k poles. To determine these zeros, in order to estimate the characteristic value, we must assign a particular value of x
in the Neumann series and this is usually done using the principle of minimal sensitivity [1, 2].
Issues of existence, uniqueness and other related de nitions of classical Pade approximants are
treated in [4, 5] and many other texts.
R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
293
3. The functional Pade approximants method (FPA)
We de ne a rational function r(x; ) to be a functional Pade approximant of type (n; 2k) for f(x; )
if
r(x; ) = p(x; )=q();
(21)
where p(x; ); q() are polynomials in ; p(x; ) ∈ L2 [a; b] as a function of x and
@{p} 6n −
@{q} 62k − 2
)
for ¿0;
(22a)
Z
2
b
q() p(x; ) dx;
a
(22b)
q() = q∗ ();
(22c)
q(0) 6= 0;
(22d)
p(x; ) − q() f(x; ) = 0 (n+1 ):
(22e)
The asterisk in (22c) denotes the functional complex conjugate.
If p(x; ), q(), satisfy (22a)–(e), then r(x; ) de ned by (1) is unique; the questions of existence,
uniqueness and degeneracy are treated in [9]. The explicit formula for the denominator polynomial
is given by
0
M01
−M01
0
..
.
..
q() =
.
−M
−M
0; 2k−1
1; 2k−1
2k
2k−1
· · · M0; 2k−1 M0; 2k
· · · M1; 2k−1 M1; 2k
..
..
..
:
.
.
.
···
0
M2k−1; 2k
···
1
(23)
The elements of (23) are de ned by
Mij =
j−i−1 Z b
X
i=0
a
Cl+i+n−2k+1 (x)[Cj−l+n−2k (x)]∗ dx;
(24)
for i = 0; 1; : : : ; 2k and j = i + 1; i + 2; : : : ; 2k and taking Cj (x) := 0 if j ¡ 0:
We know that the polynomials produced by (23) are strictly positive for ∈ R and their zeros
occur in complex conjugate pairs close to the real axis [6, 7, 11]. Ideally, we take the real parts of
the zeros of q() as our estimates of the characteristic values c .
The numerator polynomial p(x; ) follows from (22e) as
p(x; ) = [q() f(x; )]n0 :
(25)
If, in the representation (23), q(0) 6= 0, then r(x; ) de ned by (21), (23) and (25) is the functional
Pade approximant of type (n; 2k) for f(x; ).
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R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
In addition, the denominator polynomial of a hybrid functional Pade approximant is de ned in
terms of the roots of the denominator of the corresponding functional Pade approximant [6, 7, 11].
We actually take the real parts of the roots of the functional Pade approximant and express the
hybrid functional Pade approximant of type (n; k) as
qH () =
k
Y
i=1
( − iR ):
(26)
The associated numerator polynomial is de ned as
pH (x; ) = [f(x; ) qH ()]n0 :
(27)
Since n and k govern the degree of the numerator and denominator, respectively, we express the
hybrid functional Pade approximant of type (n; k) as
r H (x; ) = pH (x; )=qH ():
(28)
We nd that (28) is identical to the integral Pade approximant (2) for k = 1. It is obvious that
the hybrid functional Pade approximant is dependent on functional Pade approximant and therefore
requires much more numerical computation than the integral Pade approximant. We can easily prove
this by comparing the dimensions of the determinant and the degree of the denominator polynomial
of functional Pade approximant and integral Pade approximant, which are given by (23) and (4),
respectively.
4. Equivalence of the estimates
Here we shall observe how three methods, namely integral Pade approximant, the squared Pade approximant and the functional Pade approximant produce similar estimates of the characteristic value.
We shall do this by showing that the limit of the poles of the respective denominator polynomial
satis es identical equations.
First we begin by expanding (4), the denominator polynomial of the integral Pade approximant
of type (n; 1):
Z b
C (x) Cn−1 (x) dx
D() = a n
Z
b
Cn2 (x) dx
a
1
which gives
D() =
Z
a
b
Cn (x) Cn−1 (x) dx −
Z
a
;
b
Cn2 (x) dx:
(29)
The characteristic value of the integral Pade approximant of type (n; 1) is calculated by solving (29)
and thus we have
=
Z
a
b
Cn (x)Cn−1 (x) dx
,Z
a
b
Cn2 (x) dx:
(30)
R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
295
Similarly, expanding the denominator polynomial of the functional Pade approximant of type (n; 2),
0
Z b
q() =
2
(x) dx
Cn−1
−
a
2
b
Z
2
(x) dx
Cn−1
a
2
b
Z
Cn−1 (x)Cn (x) dx
a
Z
0
a
b
;
Cn2 (x) dx
1
which gives
Z
q() =
b
2
Cn−1
(x) dx
a
−2
Z
a
"Z
b
2
Cn−1
(x) dx
a
b
2
(x) dx
Cn−1
"Z
+
2
b
Z
a
Cn2 (x) dx
#
#
b
Cn (x)Cn−1 (x) dx :
a
(31)
Solving (31) by a standard quadratic formula and simplifying gives rise to
=
Rb
[ a Cn (x)Cn−1 (x) dx ±
q R
b
[{ a Cn (x)Cn−1 (x) dx}2 −
Rb
[ a Cn2 (x) dx]
Rb
a
Rb
2
Cn2 (x) dx a Cn−1
(x) dx]]
:
(32)
We know from previous studies [6, 7, 11] that (32) produces a pair of complex conjugates which
are close to the real axis, that is
"Z
b
Cn (x)Cn−1 (x) dx
a
#2
¡
"Z
b
a
Cn2 (x) dx
Z
a
b
2
Cn−1
(x) dx
#
(33)
and
lim
n→∞
"Z
b
Cn (x)Cn−1 (x) dx
a
#2
"Z
=
a
b
Cn2 (x) dx
Z
a
b
2
(x) dx
Cn−1
#
:
(34)
Therefore, the estimate of the characteristic value of the functional Pade approximant of type (n; 2)
method is given by
=
"Z
a
b
Cn (x)Cn−1 (x) dx
# ,"Z
a
b
#
Cn2 (x) dx :
(35)
Alternatively, we can nd the estimates of the functional Pade approximant by di erentiating (31)
w.r.t. and with a convenient normalisation we obtain (35) [8, 11].
First we begin by expanding (15), the denominator polynomial of the squared Pade approximant
of type (n; 1)
Z b
Z b
2
2
(x)
dx
C
(x)
dx
C
n
n−1
;
H () = a
a
1
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R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
which gives
Z
H () =
b
2
(x) dx
Cn−1
a
−
Z
a
b
Cn2 (x) dx:
(36)
The characteristic value of squared Pade approximant of type (n; 1) is based on solving (36); thus
we have
=
Z
b
2
(x) dx
Cn−1
a
,Z
a
b
Cn2 (x) dx:
(37)
If we rearrange (34), we have
lim
n→∞
"Z
a
b
2
(x) dx
Cn−1
#
=
Rb
[ a Cn (x)Cn−1 (x) dx]2
Rb
[ a Cn2 (x) dx]
(38)
and substituting the right-hand side of (38) into (37) we obtain
=
"R b
a
Cn (x)Cn−1 (x) dx
Rb
a
Cn2 (x) dx
#2
:
(39)
It has been established in [5, 6, 8, 10] that the zeros, that is characteristic value, of the functional
Pade approximant converges as
lim n = 2 :
n →∞
(40)
This result also applies to the squared Pade approximant.
However, we actually take the square root of (37) as our estimates of the characteristic value
v
,Z
u Z b
b
u
2
t
= Cn−1 (x) dx
Cn2 (x) dx :
a
a
(41)
Therefore, we expect that, for all n suciently large, (41) is equivalent to (30). It should be
noted that this is true mathematically, but we nd a serious problem numerically that is when the
generating function possesses an alternating or a negative power series.
Mathematically, we have demonstrated the fact that these three methods produce a similar accuracy
for the estimates of the characteristic value, that is (30), (35) and (41) are similar for the rst row
sequence. We conjecture that these three methods produce identical estimates for any type of row
sequence. However, numerically we nd that the integral Pade approximant has better precision than
the squared Pade approximant and the functional Pade approximant. This is empirically evident from
Table 2.
5. Application to an integral equation
To determine the consistency of these new methods, we actually tested them on a previous
investigation [11] and further examples were taken from Moiseiwitsch [12]. These ndings are
R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
297
generalised by illustrating the e ectiveness of these new methods for determining the characteristic
values and the characteristic function of a familiar linear integral equation in the following example.
We investigate the convergence of sequences of these new methods for the Neumann series
solution of the linear integral equation
f(x; ) = 1 +
Z
1
k(x; y)f(y; ) dy;
(42)
0
where
k(x; y) =
1 + x − y;
1 + y − x;
0 6 y 6 x 6 1;
0 6 x 6 y 6 1:
This integral equation is a Fredholm of the second kind with a nondegenerate kernel and has been
previously considered by Graves–Morris [7] and Coope and Graves–Morris [6].
The characteristic functions of this equation can be found by converting it to a second-order
ordinary di erential equation [3]. The explicit solution of (42) is
2 cosh (x − 1=2)
f(x; ) =
;
(43)
2 cosh =2 − 3 sinh =2
√
where = 2. The denominator of (43) is analytic as a function of and has just one simple zero
at c = 1:22290658 : : :, corresponding to a single characteristic value c = 0:7477502556 : : :.
It is familiar that the Neumann series of (42) converges for ||¡c [12]. The rst few terms of
this series are
"
"
#
#
∞
X
161 5
5
1 2 1
1 4 2
1 2
i
f(x; ) =
x−
x−
+
+ ···;
+
Ci (x) = 1 +
+ x−
+
4
2
96
4
2
6
2
i=0
(44)
as may be found by iteration of (42).
We make two distinct comparisons of the estimates derived using the integral Pade approximants.
First, we compare estimates formed using the row sequence of the integral Pade approximant of
type (n; 1) with corresponding estimates derived from the classical Pade approximant of type (n; 1),
the modi ed Pade approximant of type (n; 1), the squared Pade approximant of type (n; 1) and
the functional Pade approximant of type (n; 2). The results in Table 1 are the estimates of the
characteristic value for each of the ve iterative methods described and we nd that the estimates
from the integral Pade approximant gives better approximations that the classical Pade approximant,
the modi ed Pade approximant and it is similar to the functional Pade approximant and the squared
Pade approximant. The results in Table 2 are the estimates of the characteristic value, but showing
results derived from another row sequence of the integral Pade approximant of type (n; 2) and with
corresponding estimates derived from the classical Pade approximant of type (n; 2), the modi ed
Pade approximant of type (n; 2), the squared Pade approximant of type (n; 2) and the functional
Pade approximant of type (n; 4). We see that the row sequence of the integral Pade approximants
gives better approximation than the other methods, including the functional Pade approximants. In
each case, the comparisons with other methods were made using a similar amount of data, which
is, using a similar number of terms of (44).
The proof that the row sequence of Pade approximant estimates shown in Table 1 converge geometrically to the characteristic value follows as a consequence of Sa ’s [14] extension of Montessus’
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R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
Table 1
Estimates showing the precision of the characteristic value c derived using the ve methods described
n
IPA = FPA
ca
SPA
c
MPA
c
CPA
c
1
2
3
4
5
0.74766
0.74775011
0.7477502553
0.7477502555635
0.7477502555638444
0.7488
0.747752
0.7477502583
0.747750255568
0.747750255563853
0.75
0.74766
0.747754
0.74775011
0.747750261
0.8
0.745
0.74785
0.747746
0.74775043
a
The exact value of c = 0:747750255563845043 : : : .
Table 2
Estimates showing the precision of the characteristic value c derived using the ve methods described
IPA
n ca
FPA
c
SPA
c
MPA
c
CPA
c
1
2
3
4
0.7453
0.74775025556262
0.74775025556384496
0.7477502555638450433815
0.747755
0.7477502557
0.747750255563858
0.747750255563845045
0.74767
0.7477504
0.7477502545
0.74775025557
0.7486
0.74774
0.7477504
0.74775025
0.7475011
0.74775025556375
0.747750255563845035
0.7477502555638450433881
a
The exact value of c = 0:74775025556384504338894 : : : .
theorem [13]. Likewise, the proof that the functional Pade approximant estimates shown in the Table
1 converge geometrically to the characteristic value, follows as a consequence of the row convergence theorem of Graves–Morris and Sa [10]. All these mathematical results are empirically evident
from Tables 1 and 2. It is also clear that the integral Pade approximant method converges much
faster than the other methods. This is due to the standard theory of Fredholm integral equations of
the second kind [12, 15], that is expressed by
f(x; ) = g(x) +
Z
b
k(x; y)f(y; ) dy;
(45)
a
it is known that when g(x) ∈ L2 [a; b] and k(x; y) is an L2 kernel,
f(x; y) =
P(x; )
;
Q()
(46)
where
(i) P(x; ) is holomorphic as a function of ,
(ii) P(x; ) ∈ L2 [a; b] for a xed ,
(iii) Q() is holomorphic function of ,
(iv) Q(0) 6= 0.
We have noted that the more serious practical diculty with the classical Pade approximant method
is that the estimates of the characteristic values of depend on the value of x selected contrary
to the exact result (46). If we use the integral Pade approximants for accelerating convergence of
the Neumann series, the estimates of the characteristic values are independent of the variable x,
as expressed in (46). This observation goes some way towards explaining the remarkable precision
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R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
of estimates of the characteristic values derived from the integral Pade approximants as shown in
Tables 1 and 2.
6. Precision of the approximate solution
In Fig. 1 we display the exact (analytic) solution and its approximations obtained using the integral Pade approximant of type (2; 1), the modi ed Pade approximant of type (2; 1), the squared
Pade approximant of type (2; 1) and the functional Pade approximant of type (2; 2). Also in
Fig. 1 we see a remarkable precision of the integral Pade approximant, where graphically there
is no signi cant di erence between the exact and the integral Pade approximant. Accordingly, in
Table 3, we show the errors incurred by the integral Pade approximant, the modi ed Pade approximant, the squared Pade approximant and the functional Pade approximant methods for x = 0(0:1)0:5
in the solution of (42). We list the appropriate rational functions displayed.
Solution of the integral equation (42) using the integral Pade approximant of type (2; 1) is
N (x; ) 1 + x2 − x +
r(x; ) =
=
D()
4399
27045
+
1 4
x
6
− 13 x3 +
72337
1 − 54090
4399 2
x
27045
+
217
x
54090
−
217
36060
2
:
(47)
Fig. 1. The analytic solution (exact) of (42) for = 1. The curves IPA and MPA are indistinguishable from the exact
curve. Whereas the curve FPA is perceptibly di erent and SPA is beyond the exact curve.
Table 3
Errors occurring in the solution of (42) using the integral Pade approximant, the modi ed Pade approximant, the squared
Pade approximant and the functional Pade approximant method
x
IPA
=1
MPA
=1
SPA
=1
FPA
=1
0
0.1
0.2
0.3
0.4
0.5
−0.00088
−0.00066
−0.00021
0.00031
0.00070
0.00085
−0.0036
−0.0032
−0.0026
−0.0020
−0.0016
−0.0014
−3.39
−3.19
−3.04
−2.93
−2.86
−2.84
0.069
0.050
0.013
−0.026
−0.053
−0.063
300
R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
Solution of (42) based on the squared Pade approximant of type (2; 1) is
G(x; ) 1 + x2 − x +
=
r(x; ) =
H ()
2917
101115
+
1 4
x
6
− 13 x3 −
72337
1 − 40446
2917 2
x
101115
+
18409
x
40446
−
18409
26964
2
:
(48)
Solution of (42) based on the functional Pade approximant of type (2; 2) is
p(x; ) 1 + x2 − x −
=
r(x; ) =
q()
5279
4494
x2 −
+ 16 x4 − 13 x3 − 5279
4494
1 − 18030
+ 72337
2
6741
40446
1507
x
40446
−
9041
40446
2
:
(49)
Solution of (42) based on the modi ed Pade approximant of type (2; 1) is
A(x; ) 1 + x2 − x +
=
r(x; ) =
B()
13
80
+
1 4
x
6
− 13 x3 +
1 − 107
80
13 2
x
80
+
1
x
15
−
1
10
2
:
(50)
The rational functions above were suitably normalised.
It has been established in previous studies [6, 7, 11] that the estimates of characteristic function
based on the functional Pade approximants are inferior to those from the hybrid functional Pade
approximants. The reason for the poor performance of the functional Pade approximants for the
estimation of the characteristic function is that the denominator polynomials (11) possess super
uous
zeros, and thus the hybrid functional Pade approximant method was introduced. We do not use the
hybrid functional Pade approximant, not because it produces identical results to the integral Pade
approximant, but because of much more numerical computation involved in deriving it. Furthermore,
we have found that the precision of the characteristic function for the squared Pade approximant
method is considerably smaller than the integral Pade approximant. Finally, although the approximate
solution for the characteristic function for the modi ed Pade approximant is satisfactory, it lacks the
precision of the integral Pade approximant.
7. Remarks and conclusion
Three new methods for producing a sequence of rational approximations to the solution of a linear
integral equation have been described and their e ectiveness has been investigated in many examples.
These new methods are essentially for accelerating the convergence of a sequence of functions. In
the context of a familiar linear integral equation, the method of integral Pade approximants is shown
to be much more ecient in calculating the characteristic value and substantially more accurate for
calculating the approximate solution than other similar techniques.
The purpose of demonstrating the integral Pade approximants method for two types of row sequence is to illustrate the accuracy of the approximate solution, the stability of the convergence and
the consistency of the method. Furthermore, we have demonstrated that this new method is much
more ecient and does not have the drawbacks of the functional Pade approximants and the classical Pade approximants. From this illustrated example and in all other test examples, it is clear that
the hybrid functional Pade approximants method is inferior to the integral Pade approximants for
k ¿ 2, because of the accuracy of the characteristic values of the functional Pade approximants is
less than the integral Pade approximants method. Finally, an analytical investigation of the integral
Pade approximant is a subject of further research.
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R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
Acknowledgements
I am greatly indebted to an anonymous referee and Professor D.B. Ingham for helpful comments
on this paper.
Appendix. Application to another integral equation
We illustrate the e ectiveness of new methods by taking another well known example. We investigate the convergence of sequences of integral Pade approximants for the Neumann series solution
of the linear integral equation
f(x; ) = 1 +
Z
1
k(x; y)f(y) dy;
(A.1)
−1
where
k(x; y) =
(
8−1 (1 + y)(1 − x); −16y6x61;
8−1 (1 + x)(1 − y); −16x6y61:
This integral equation is a Fredholm of the secod kind with a nondegenerate kernel previously
considered by Graves-Morris and Thukral [11].
The characteristic functions of this equation can be found by converting it to a second-order
ordinary di erential equation [3]. The explicit solution of (1) is
f(x; s ) = sin[2−1 s(1 + x)];
s ∈ ℵ;
Table 4
Estimates showing the precision of the characteristic value 1 derived using the ve methods described
n
IPA= FPA
1a
SPA
1
MPA
1
CPA
1
1
2
3
4
10
9.871
9.86962
9.8696046
10.95
9.877
9.86969
9.8696054
12
10
9.88
9.871
8
9.6
9.84
9.866
a
The exact value of 1 = 2 = 9:8696044010 : : : .
Table 5
Estimates showing the precision of the characteristic value 1 derived using the ve methods described
n
IPA
1a
FPA
1
SPA
1
MPA
1
CPA
1
1
2
3
4
5
9.871
9.8696046
9.8696044013
9.8696044010896
9.8696044010893593
9.24
9.869609
9.8696044025
9.869604401091
9.869604401089361
10.03
9.86964
9.86960444
9.86960440115
9.86960440108946
10.25
9.8751
9.86975
9.869609
9.8696046
10.14
9.878
9.8699
9.86962
9.8696051
a
The exact value of 1 = 2 = 9:8696044010893586188 : : : .
302
R. Thukral / Journal of Computational and Applied Mathematics 102 (1999) 287–302
Table 6
Estimates showing the precision of the characteristic value 3 derived using the three methods described
n
IPA
3a
FPA
3
SPA
3
MPA
3
CPA
3
1
2
3
4
5
—
90.1
88.97
88.845
88.8289
—
104.5
90.26
88.999
88.848
—
126.1
90.99
89.07
88.858
—
170.1
102.1
92.6
90.1
—
64.79
78.31
84.52
87.15
a
The exact value of 3 = 92 = 88:8264396 : : : .
with corresponding characteristic value
s = (s)2 ;
s ∈ ℵ:
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